Abstract
We outline basics of a new approach to transverse momentum dependence in hard processes. As an illustration, we consider hard exclusive transition process γ⁎γ→π0 at the handbag level. Our starting point is coordinate representation for matrix elements of operators (in the simplest case, bilocal O(0,z)) describing a hadron with momentum p. Treated as functions of (pz) and z2, they are parametrized through virtuality distribution amplitudes (VDA) Φ(x,σ), with x being Fourier-conjugate to (pz) and σ Laplace-conjugate to z2. For intervals with z+=0, we introduce the transverse momentum distribution amplitude (TMDA) Ψ(x,k⊥), and write it in terms of VDA Φ(x,σ). The results of covariant calculations, written in terms of Φ(x,σ) are converted into expressions involving Ψ(x,k⊥). Starting with scalar toy models, we extend the analysis onto the case of spin-1/2 quarks and QCD. We propose simple models for soft VDAs/TMDAs, and use them for comparison of handbag results with experimental (BaBar and BELLE) data on the pion transition form factor. We also discuss how one can generate high-k⊥ tails from primordial soft distributions.
Highlights
Analysis of effects due to parton transverse momentum is an important direction in modern studies of hadronic structure
In case of spin-1/2 quarks interacting via ascalar gluon field (“Yukawa” gluon model), Eq (67) is modified by an extra k⊥2 factor coming from the numerator spinor trace, which leads to 1/(k⊥2 + m2) dependence in the correction (64) to the transverse momentum dependent distribution amplitude (TMDA)
The structure of a hadron with momentum p is described by a matrix element of the bilocal operator O(0, z), treated as a function of and z2
Summary
Analysis of effects due to parton transverse momentum is an important direction in modern studies of hadronic structure. In our OPE-type analysis of the γ∗γ → π0 process performed in the present paper, we encounter a TMD-like object, the transverse momentum dependent distribution amplitude (TMDA) Ψ(x, k⊥) that is a 3-dimensional generalization of the pion distribution amplitude φπ(x) [9, 10, 11, 12]. We discuss the structure of the relevant bilocal matrix element p|φ(0)φ(z)|0 as a function of Lorentz invariants (pz) and z2 and introduce the virtuality distribution amplitude (VDA) Φ(x, σ), the basic object of our approach. It describes the distribution of quarks in the pion both in the longitudinal momentum (the variable x is conjugate to (pz)) and in virtuality (the variable σ is conjugate to z2).
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